Abstract
Field trials for variety selection often exhibit spatial correlation between plots. When multivariate data are analysed from these field trials, there is the added complication in having to simultaneously account for correlation between the traits at both the residual and genetic levels. This may be temporal correlation in the case of multi-harvest data from perennial crop field trials, or between-trait correlation in multi-trait data sets. Use of parsimonious yet plausible models for the variance–covariance structure of the residuals for such data is a key element to achieving an efficient and inferentially sound analysis. In this paper, a model is developed for the residual variance–covariance structure firstly by considering a multivariate autoregressive model in one spatial direction and then extending this to two spatial directions. Conditions for ensuring that the processes are directionally invariant are presented. Using a canonical decomposition, these directionally invariant processes can be transformed into a set of independent separable processes. This simplifies the estimation process. The new model allows for flexible modelling of the spatial and multivariate interaction and allows for different spatial correlation parameters for each harvest or trait. The methods are illustrated using data from lucerne breeding trials at several environments.
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Acknowledgements
The authors thank Shoba Venkatanagappa and the New South Wales Department of Primary Industries lucerne breeding team for the motivating dataset. The first author gratefully acknowledges a postgraduate research scholarship contributed by the New South Wales Department of Primary Industries and Fisheries and supplementary scholarships by the Division of Mathematical and Information Sciences, CSIRO, and the School of Agriculture, Food and Wine, The University of Adelaide. The authors acknowledge the financial support of Grains Research and Development Corporation and New South Wales Department of Primary Industries for the lucerne breeding programme. The authors also thank Arthur Gilmour and David Butler for support with software development. The last author acknowledges the insightful comments of Beverley Gogel. The authors also thank one reviewer for suggestions which greatly simplified the derivations and the other reviewer for suggestions that led to improvements in the example.
AUTHOR’S CONTRIBUTIONS
The work in this paper is based on the PhD thesis of JD which was supervised by BC and WP with technical guidance provided by AV. The PhD was instigated by BC and AV with funding found by BC and AV. Later, the canonical decomposition and 2D simultaneous autoregressive process were suggested by RT. JD and AV wrote the paper and all authors were involved in reviewing the manuscript. JD wrote the required code and analysed the data.
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APPENDIX
APPENDIX
Let \(\varvec{A}\) and \(\varvec{B}\) be two matrices partitioned as
where the partitioned matrices are all \(t\times t\). A generalized product of \(\varvec{A}\) and \(\varvec{B}\) is defined as
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De Faveri, J., Verbyla, A.P., Cullis, B.R. et al. Residual Variance–Covariance Modelling in Analysis of Multivariate Data from Variety Selection Trials. JABES 22, 1–22 (2017). https://doi.org/10.1007/s13253-016-0267-0
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DOI: https://doi.org/10.1007/s13253-016-0267-0